Question

A park is to be designed as a circle. A straight walkway will intersect the fence of the

park twice, requiring gates at each location. The city planner draws the circular park
and the walkway on a coordinate plane, with the equation
x² + y² - 4x = 9 for the circular park and the equation y = 2x modeling the
walkway. Write an ordered pair that represents the location of the gates in the third
quadrant.

Answer 1

o find the coordinates of the gates in the third quadrant, we need to find the points where the circle and the line intersect in the third quadrant.

Substituting y = 2x into x² + y² - 4x = 9, we get:

x² + (2x)² - 4x = 9

5x² - 4x - 9 = 0

Using the quadratic formula, we find:

x = (-(-4) ± √((-4)² - 4(5)(-9))) / (2(5))

x = (4 ± √136) / 10

We can discard the positive root since it is in the first quadrant. The negative root corresponds to the x-coordinate of the point of intersection in the third quadrant:

x = (4 - √136) / 10 ≈ -0.433

Substituting this value into y = 2x, we get:

y = 2(-0.433) ≈ -0.866

Therefore, the ordered pair that represents the location of the gates in the third quadrant is (-0.433, -0.866).